ITP001 Axioms: ITP089^7.ax


%------------------------------------------------------------------------------
% File     : ITP089^7 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 syntactic export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : toto.ax [Gau19]
%          : HL4089^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  225 (  50 unt;  94 typ;   0 def)
%            Number of atoms       :  421 ( 239 equ;  19 cnn)
%            Maximal formula atoms :   18 (   1 avg)
%            Number of connectives : 2002 (  19   ~;   8   |; 110   &;1755   @)
%                                         (  34 <=>;  76  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   28 (   7 avg;1755 nst)
%            Number of types       :    6 (   5 usr)
%            Number of type conns  :  367 ( 367   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   91 (  89 usr;  13 con; 0-6 aty)
%            Number of variables   :  568 (  19   ^ 476   !;  12   ?; 568   :)
%                                         (  61  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Eind__type_2Erecspace,type,
    tyop_2Eind__type_2Erecspace: $tType > $tType ).

thf(tyop_2Elist_2Elist,type,
    tyop_2Elist_2Elist: $tType > $tType ).

thf(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: $tType ).

thf(tyop_2Emin_2Efun,type,
    tyop_2Emin_2Efun: $tType > $tType > $tType ).

thf(tyop_2Enum_2Enum,type,
    tyop_2Enum_2Enum: $tType ).

thf(tyop_2Epair_2Eprod,type,
    tyop_2Epair_2Eprod: $tType > $tType > $tType ).

thf(tyop_2Estring_2Echar,type,
    tyop_2Estring_2Echar: $tType ).

thf(tyop_2EternaryComparisons_2Eordering,type,
    tyop_2EternaryComparisons_2Eordering: $tType ).

thf(tyop_2Etoto_2Enum__dt,type,
    tyop_2Etoto_2Enum__dt: $tType ).

thf(tyop_2Etoto_2Etoto,type,
    tyop_2Etoto_2Etoto: $tType > $tType ).

thf(c_2Ebool_2E_21,type,
    c_2Ebool_2E_21: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Earithmetic_2E_2B,type,
    c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Epair_2E_2C,type,
    c_2Epair_2E_2C: 
      !>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).

thf(c_2Earithmetic_2E_2D,type,
    c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Ebool_2E_2F_5C,type,
    c_2Ebool_2E_2F_5C: $o > $o > $o ).

thf(c_2Enum_2E0,type,
    c_2Enum_2E0: tyop_2Enum_2Enum ).

thf(c_2Eprim__rec_2E_3C,type,
    c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).

thf(c_2Emin_2E_3D,type,
    c_2Emin_2E_3D: 
      !>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).

thf(c_2Emin_2E_3D_3D_3E,type,
    c_2Emin_2E_3D_3D_3E: $o > $o > $o ).

thf(c_2Ebool_2E_3F,type,
    c_2Ebool_2E_3F: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Emin_2E_40,type,
    c_2Emin_2E_40: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).

thf(c_2Ebool_2EARB,type,
    c_2Ebool_2EARB: 
      !>[A_27a: $tType] : A_27a ).

thf(c_2Earithmetic_2EBIT1,type,
    c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Earithmetic_2EBIT2,type,
    c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Eind__type_2EBOTTOM,type,
    c_2Eind__type_2EBOTTOM: 
      !>[A_27a: $tType] : ( tyop_2Eind__type_2Erecspace @ A_27a ) ).

thf(c_2Estring_2ECHR,type,
    c_2Estring_2ECHR: tyop_2Enum_2Enum > tyop_2Estring_2Echar ).

thf(c_2Ebool_2ECOND,type,
    c_2Ebool_2ECOND: 
      !>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).

thf(c_2Elist_2ECONS,type,
    c_2Elist_2ECONS: 
      !>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).

thf(c_2Eind__type_2ECONSTR,type,
    c_2Eind__type_2ECONSTR: 
      !>[A_27a: $tType] : ( tyop_2Enum_2Enum > A_27a > ( tyop_2Enum_2Enum > ( tyop_2Eind__type_2Erecspace @ A_27a ) ) > ( tyop_2Eind__type_2Erecspace @ A_27a ) ) ).

thf(c_2Ebool_2EDATATYPE,type,
    c_2Ebool_2EDATATYPE: 
      !>[A_27a: $tType] : ( A_27a > $o ) ).

thf(c_2Earithmetic_2EDIV2,type,
    c_2Earithmetic_2EDIV2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2EternaryComparisons_2EEQUAL,type,
    c_2EternaryComparisons_2EEQUAL: tyop_2EternaryComparisons_2Eordering ).

thf(c_2Ebool_2EF,type,
    c_2Ebool_2EF: $o ).

thf(c_2Eind__type_2EFCONS,type,
    c_2Eind__type_2EFCONS: 
      !>[A_27a: $tType] : ( A_27a > ( tyop_2Enum_2Enum > A_27a ) > tyop_2Enum_2Enum > A_27a ) ).

thf(c_2Epair_2EFST,type,
    c_2Epair_2EFST: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).

thf(c_2EternaryComparisons_2EGREATER,type,
    c_2EternaryComparisons_2EGREATER: tyop_2EternaryComparisons_2Eordering ).

thf(c_2Ecombin_2EI,type,
    c_2Ecombin_2EI: 
      !>[A_27a: $tType] : ( A_27a > A_27a ) ).

thf(c_2EternaryComparisons_2ELESS,type,
    c_2EternaryComparisons_2ELESS: tyop_2EternaryComparisons_2Eordering ).

thf(c_2Epair_2ELEX,type,
    c_2Epair_2ELEX: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27b > A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).

thf(c_2Erelation_2ELinearOrder,type,
    c_2Erelation_2ELinearOrder: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).

thf(c_2Etoto_2EListOrd,type,
    c_2Etoto_2EListOrd: 
      !>[A_27a: $tType] : ( ( tyop_2Etoto_2Etoto @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > tyop_2EternaryComparisons_2Eordering ) ).

thf(c_2Elist_2ENIL,type,
    c_2Elist_2ENIL: 
      !>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).

thf(c_2Earithmetic_2ENUMERAL,type,
    c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Earithmetic_2EODD,type,
    c_2Earithmetic_2EODD: tyop_2Enum_2Enum > $o ).

thf(c_2Ebool_2EONE__ONE,type,
    c_2Ebool_2EONE__ONE: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > $o ) ).

thf(c_2Estring_2EORD,type,
    c_2Estring_2EORD: tyop_2Estring_2Echar > tyop_2Enum_2Enum ).

thf(c_2Epair_2ESND,type,
    c_2Epair_2ESND: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).

thf(c_2Erelation_2ESTRORD,type,
    c_2Erelation_2ESTRORD: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).

thf(c_2Enum_2ESUC,type,
    c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Erelation_2EStrongLinearOrder,type,
    c_2Erelation_2EStrongLinearOrder: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).

thf(c_2Etoto_2EStrongLinearOrder__of__TO,type,
    c_2Etoto_2EStrongLinearOrder__of__TO: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > A_27a > A_27a > $o ) ).

thf(c_2Erelation_2EStrongOrder,type,
    c_2Erelation_2EStrongOrder: 
      !>[A_27g: $tType] : ( ( A_27g > A_27g > $o ) > $o ) ).

thf(c_2Ebool_2ET,type,
    c_2Ebool_2ET: $o ).

thf(c_2Etoto_2ETO,type,
    c_2Etoto_2ETO: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > ( tyop_2Etoto_2Etoto @ A_27a ) ) ).

thf(c_2Etoto_2ETO__inv,type,
    c_2Etoto_2ETO__inv: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) ).

thf(c_2Etoto_2ETO__of__LinearOrder,type,
    c_2Etoto_2ETO__of__LinearOrder: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) ).

thf(c_2Ebool_2ETYPE__DEFINITION,type,
    c_2Ebool_2ETYPE__DEFINITION: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).

thf(c_2Etoto_2ETotOrd,type,
    c_2Etoto_2ETotOrd: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > $o ) ).

thf(c_2Epair_2EUNCURRY,type,
    c_2Epair_2EUNCURRY: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).

thf(c_2Erelation_2EWF,type,
    c_2Erelation_2EWF: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).

thf(c_2Erelation_2EWFREC,type,
    c_2Erelation_2EWFREC: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( ( A_27a > A_27b ) > A_27a > A_27b ) > A_27a > A_27b ) ).

thf(c_2Erelation_2EWeakLinearOrder,type,
    c_2Erelation_2EWeakLinearOrder: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).

thf(c_2Etoto_2EWeakLinearOrder__of__TO,type,
    c_2Etoto_2EWeakLinearOrder__of__TO: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > A_27a > A_27a > $o ) ).

thf(c_2Earithmetic_2EZERO,type,
    c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).

thf(c_2Ebool_2E_5C_2F,type,
    c_2Ebool_2E_5C_2F: $o > $o > $o ).

thf(c_2Etoto_2Eapto,type,
    c_2Etoto_2Eapto: 
      !>[A_27a: $tType] : ( ( tyop_2Etoto_2Etoto @ A_27a ) > A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) ).

thf(c_2Etoto_2Ebit1,type,
    c_2Etoto_2Ebit1: tyop_2Etoto_2Enum__dt > tyop_2Etoto_2Enum__dt ).

thf(c_2Etoto_2Ebit2,type,
    c_2Etoto_2Ebit2: tyop_2Etoto_2Enum__dt > tyop_2Etoto_2Enum__dt ).

thf(c_2Etoto_2EcharOrd,type,
    c_2Etoto_2EcharOrd: tyop_2Estring_2Echar > tyop_2Estring_2Echar > tyop_2EternaryComparisons_2Eordering ).

thf(c_2Estring_2Echar__lt,type,
    c_2Estring_2Echar__lt: tyop_2Estring_2Echar > tyop_2Estring_2Echar > $o ).

thf(c_2Etoto_2Echarto,type,
    c_2Etoto_2Echarto: tyop_2Etoto_2Etoto @ tyop_2Estring_2Echar ).

thf(c_2Etoto_2EimageOrd,type,
    c_2Etoto_2EimageOrd: 
      !>[A_27a: $tType,A_27c: $tType] : ( ( A_27a > A_27c ) > ( A_27c > A_27c > tyop_2EternaryComparisons_2Eordering ) > A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) ).

thf(c_2Erelation_2Einv,type,
    c_2Erelation_2Einv: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > A_27b > A_27a > $o ) ).

thf(c_2Erelation_2Eirreflexive,type,
    c_2Erelation_2Eirreflexive: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).

thf(c_2Etoto_2ElexTO,type,
    c_2Etoto_2ElexTO: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > ( A_27b > A_27b > tyop_2EternaryComparisons_2Eordering ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > tyop_2EternaryComparisons_2Eordering ) ).

thf(c_2Etoto_2Elextoto,type,
    c_2Etoto_2Elextoto: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Etoto_2Etoto @ A_27a ) > ( tyop_2Etoto_2Etoto @ A_27b ) > ( tyop_2Etoto_2Etoto @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ).

thf(c_2Etoto_2Elistorder,type,
    c_2Etoto_2Elistorder: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).

thf(c_2Etoto_2Elistoto,type,
    c_2Etoto_2Elistoto: 
      !>[A_27a: $tType] : ( ( tyop_2Etoto_2Etoto @ A_27a ) > ( tyop_2Etoto_2Etoto @ ( tyop_2Elist_2Elist @ A_27a ) ) ) ).

thf(c_2Etoto_2EnumOrd,type,
    c_2Etoto_2EnumOrd: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2EternaryComparisons_2Eordering ).

thf(c_2Etoto_2Enum__dtOrd,type,
    c_2Etoto_2Enum__dtOrd: tyop_2Etoto_2Enum__dt > tyop_2Etoto_2Enum__dt > tyop_2EternaryComparisons_2Eordering ).

thf(c_2Etoto_2Enum__dt__CASE,type,
    c_2Etoto_2Enum__dt__CASE: 
      !>[A_27a: $tType] : ( tyop_2Etoto_2Enum__dt > A_27a > ( tyop_2Etoto_2Enum__dt > A_27a ) > ( tyop_2Etoto_2Enum__dt > A_27a ) > A_27a ) ).

thf(c_2Etoto_2Enum__dt__size,type,
    c_2Etoto_2Enum__dt__size: tyop_2Etoto_2Enum__dt > tyop_2Enum_2Enum ).

thf(c_2Etoto_2Enum__to__dt,type,
    c_2Etoto_2Enum__to__dt: tyop_2Enum_2Enum > tyop_2Etoto_2Enum__dt ).

thf(c_2Etoto_2Enumto,type,
    c_2Etoto_2Enumto: tyop_2Etoto_2Etoto @ tyop_2Enum_2Enum ).

thf(c_2EternaryComparisons_2Eordering__CASE,type,
    c_2EternaryComparisons_2Eordering__CASE: 
      !>[A_27a: $tType] : ( tyop_2EternaryComparisons_2Eordering > A_27a > A_27a > A_27a > A_27a ) ).

thf(c_2Etoto_2Eqk__numOrd,type,
    c_2Etoto_2Eqk__numOrd: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2EternaryComparisons_2Eordering ).

thf(c_2Etoto_2Eqk__numto,type,
    c_2Etoto_2Eqk__numto: tyop_2Etoto_2Etoto @ tyop_2Enum_2Enum ).

thf(c_2Etoto_2Estringto,type,
    c_2Etoto_2Estringto: tyop_2Etoto_2Etoto @ ( tyop_2Elist_2Elist @ tyop_2Estring_2Echar ) ).

thf(c_2Etoto_2Etoto__inv,type,
    c_2Etoto_2Etoto__inv: 
      !>[A_27a: $tType] : ( ( tyop_2Etoto_2Etoto @ A_27a ) > ( tyop_2Etoto_2Etoto @ A_27a ) ) ).

thf(c_2Etoto_2Etoto__of__LinearOrder,type,
    c_2Etoto_2Etoto__of__LinearOrder: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( tyop_2Etoto_2Etoto @ A_27a ) ) ).

thf(c_2Erelation_2Etransitive,type,
    c_2Erelation_2Etransitive: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).

thf(c_2Erelation_2Etrichotomous,type,
    c_2Erelation_2Etrichotomous: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).

thf(c_2Etoto_2Ezer,type,
    c_2Etoto_2Ezer: tyop_2Etoto_2Enum__dt ).

thf(c_2Ebool_2E_7E,type,
    c_2Ebool_2E_7E: $o > $o ).

thf(logicdef_2E_2F_5C,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
    <=> ( V0
        & V1 ) ) ).

thf(logicdef_2E_5C_2F,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
    <=> ( V0
        | V1 ) ) ).

thf(logicdef_2E_7E,axiom,
    ! [V0: $o] :
      ( ( c_2Ebool_2E_7E @ V0 )
    <=> ( (~) @ V0 ) ) ).

thf(logicdef_2E_3D_3D_3E,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
    <=> ( V0
       => V1 ) ) ).

thf(logicdef_2E_3D,axiom,
    ! [A_27a: $tType,V0: A_27a,V1: A_27a] :
      ( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
    <=> ( V0 = V1 ) ) ).

thf(quantdef_2E_21,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_21 @ A_27a @ V0f )
    <=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(quantdef_2E_3F,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_3F @ A_27a @ V0f )
    <=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(thm_2Etoto_2ETotOrd,axiom,
    ! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
      ( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
    <=> ( ! [V1x: A_27a,V2y: A_27a] :
            ( ( ( V0c @ V1x @ V2y )
              = c_2EternaryComparisons_2EEQUAL )
          <=> ( V1x = V2y ) )
        & ! [V3x: A_27a,V4y: A_27a] :
            ( ( ( V0c @ V3x @ V4y )
              = c_2EternaryComparisons_2EGREATER )
          <=> ( ( V0c @ V4y @ V3x )
              = c_2EternaryComparisons_2ELESS ) )
        & ! [V5x: A_27a,V6y: A_27a,V7z: A_27a] :
            ( ( ( ( V0c @ V5x @ V6y )
                = c_2EternaryComparisons_2ELESS )
              & ( ( V0c @ V6y @ V7z )
                = c_2EternaryComparisons_2ELESS ) )
           => ( ( V0c @ V5x @ V7z )
              = c_2EternaryComparisons_2ELESS ) ) ) ) ).

thf(thm_2Etoto_2ETO__of__LinearOrder,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > $o,V1x: A_27a,V2y: A_27a] :
      ( ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r @ V1x @ V2y )
      = ( c_2Ebool_2ECOND @ tyop_2EternaryComparisons_2Eordering @ ( c_2Emin_2E_3D @ A_27a @ V1x @ V2y ) @ c_2EternaryComparisons_2EEQUAL @ ( c_2Ebool_2ECOND @ tyop_2EternaryComparisons_2Eordering @ ( V0r @ V1x @ V2y ) @ c_2EternaryComparisons_2ELESS @ c_2EternaryComparisons_2EGREATER ) ) ) ).

thf(thm_2Etoto_2Etoto__TY__DEF,axiom,
    ! [A_27a: $tType] :
    ? [V0rep: ( tyop_2Etoto_2Etoto @ A_27a ) > A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] : ( c_2Ebool_2ETYPE__DEFINITION @ ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) @ ( tyop_2Etoto_2Etoto @ A_27a ) @ ( c_2Etoto_2ETotOrd @ A_27a ) @ V0rep ) ).

thf(thm_2Etoto_2Eto__bij,axiom,
    ! [A_27a: $tType] :
      ( ! [V0a: tyop_2Etoto_2Etoto @ A_27a] :
          ( ( c_2Etoto_2ETO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0a ) )
          = V0a )
      & ! [V1r: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
          ( ( c_2Etoto_2ETotOrd @ A_27a @ V1r )
        <=> ( ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2ETO @ A_27a @ V1r ) )
            = V1r ) ) ) ).

thf(thm_2Etoto_2EWeakLinearOrder__of__TO,axiom,
    ! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1x: A_27a,V2y: A_27a] :
      ( ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a @ V0c @ V1x @ V2y )
      = ( c_2EternaryComparisons_2Eordering__CASE @ $o @ ( V0c @ V1x @ V2y ) @ c_2Ebool_2ET @ c_2Ebool_2ET @ c_2Ebool_2EF ) ) ).

thf(thm_2Etoto_2EStrongLinearOrder__of__TO,axiom,
    ! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1x: A_27a,V2y: A_27a] :
      ( ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ V0c @ V1x @ V2y )
      = ( c_2EternaryComparisons_2Eordering__CASE @ $o @ ( V0c @ V1x @ V2y ) @ c_2Ebool_2ET @ c_2Ebool_2EF @ c_2Ebool_2EF ) ) ).

thf(thm_2Etoto_2Etoto__of__LinearOrder,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
      ( ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a @ V0r )
      = ( c_2Etoto_2ETO @ A_27a @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r ) ) ) ).

thf(thm_2Etoto_2ETO__inv,axiom,
    ! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1x: A_27a,V2y: A_27a] :
      ( ( c_2Etoto_2ETO__inv @ A_27a @ V0c @ V1x @ V2y )
      = ( V0c @ V2y @ V1x ) ) ).

thf(thm_2Etoto_2Etoto__inv,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
      ( ( c_2Etoto_2Etoto__inv @ A_27a @ V0c )
      = ( c_2Etoto_2ETO @ A_27a @ ( c_2Etoto_2ETO__inv @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ) ).

thf(thm_2Etoto_2ElexTO,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1V: A_27b > A_27b > tyop_2EternaryComparisons_2Eordering] :
      ( ( c_2Etoto_2ElexTO @ A_27a @ A_27b @ V0R @ V1V )
      = ( c_2Etoto_2ETO__of__LinearOrder @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2ELEX @ A_27a @ A_27b @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ V0R ) @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27b @ V1V ) ) ) ) ).

thf(thm_2Etoto_2Elextoto,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1v: tyop_2Etoto_2Etoto @ A_27b] :
      ( ( c_2Etoto_2Elextoto @ A_27a @ A_27b @ V0c @ V1v )
      = ( c_2Etoto_2ETO @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Etoto_2ElexTO @ A_27a @ A_27b @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) @ ( c_2Etoto_2Eapto @ A_27b @ V1v ) ) ) ) ).

thf(thm_2Etoto_2EnumOrd,axiom,
    ( c_2Etoto_2EnumOrd
    = ( c_2Etoto_2ETO__of__LinearOrder @ tyop_2Enum_2Enum @ c_2Eprim__rec_2E_3C ) ) ).

thf(thm_2Etoto_2Enumto,axiom,
    ( c_2Etoto_2Enumto
    = ( c_2Etoto_2ETO @ tyop_2Enum_2Enum @ c_2Etoto_2EnumOrd ) ) ).

thf(thm_2Etoto_2Enum__dt__TY__DEF,axiom,
    ? [V0rep: tyop_2Etoto_2Enum__dt > ( tyop_2Eind__type_2Erecspace @ $o )] :
      ( c_2Ebool_2ETYPE__DEFINITION @ ( tyop_2Eind__type_2Erecspace @ $o ) @ tyop_2Etoto_2Enum__dt
      @ ^ [V1a0: tyop_2Eind__type_2Erecspace @ $o] :
          ( c_2Ebool_2E_21 @ ( ( tyop_2Eind__type_2Erecspace @ $o ) > $o )
          @ ^ [V2_27num__dt_27: ( tyop_2Eind__type_2Erecspace @ $o ) > $o] :
              ( c_2Emin_2E_3D_3D_3E
              @ ( c_2Ebool_2E_21 @ ( tyop_2Eind__type_2Erecspace @ $o )
                @ ^ [V3a0: tyop_2Eind__type_2Erecspace @ $o] :
                    ( c_2Emin_2E_3D_3D_3E
                    @ ( c_2Ebool_2E_5C_2F
                      @ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ $o ) @ V3a0
                        @ ( c_2Eind__type_2ECONSTR @ $o @ c_2Enum_2E0 @ ( c_2Ebool_2EARB @ $o )
                          @ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ $o ) ) )
                      @ ( c_2Ebool_2E_5C_2F
                        @ ( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ $o )
                          @ ^ [V5a: tyop_2Eind__type_2Erecspace @ $o] :
                              ( c_2Ebool_2E_2F_5C
                              @ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ $o ) @ V3a0
                                @ ( ^ [V6a: tyop_2Eind__type_2Erecspace @ $o] :
                                      ( c_2Eind__type_2ECONSTR @ $o @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) @ ( c_2Ebool_2EARB @ $o )
                                      @ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ $o ) @ V6a
                                        @ ^ [V7n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ $o ) ) )
                                  @ V5a ) )
                              @ ( V2_27num__dt_27 @ V5a ) ) )
                        @ ( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ $o )
                          @ ^ [V8a: tyop_2Eind__type_2Erecspace @ $o] :
                              ( c_2Ebool_2E_2F_5C
                              @ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ $o ) @ V3a0
                                @ ( ^ [V9a: tyop_2Eind__type_2Erecspace @ $o] :
                                      ( c_2Eind__type_2ECONSTR @ $o @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) @ ( c_2Ebool_2EARB @ $o )
                                      @ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ $o ) @ V9a
                                        @ ^ [V10n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ $o ) ) )
                                  @ V8a ) )
                              @ ( V2_27num__dt_27 @ V8a ) ) ) ) )
                    @ ( V2_27num__dt_27 @ V3a0 ) ) )
              @ ( V2_27num__dt_27 @ V1a0 ) ) )
      @ V0rep ) ).

thf(thm_2Etoto_2Enum__dt__case__def,axiom,
    ! [A_27a: $tType] :
      ( ! [V0v: A_27a,V1f: tyop_2Etoto_2Enum__dt > A_27a,V2f1: tyop_2Etoto_2Enum__dt > A_27a] :
          ( ( c_2Etoto_2Enum__dt__CASE @ A_27a @ c_2Etoto_2Ezer @ V0v @ V1f @ V2f1 )
          = V0v )
      & ! [V3a: tyop_2Etoto_2Enum__dt,V4v: A_27a,V5f: tyop_2Etoto_2Enum__dt > A_27a,V6f1: tyop_2Etoto_2Enum__dt > A_27a] :
          ( ( c_2Etoto_2Enum__dt__CASE @ A_27a @ ( c_2Etoto_2Ebit1 @ V3a ) @ V4v @ V5f @ V6f1 )
          = ( V5f @ V3a ) )
      & ! [V7a: tyop_2Etoto_2Enum__dt,V8v: A_27a,V9f: tyop_2Etoto_2Enum__dt > A_27a,V10f1: tyop_2Etoto_2Enum__dt > A_27a] :
          ( ( c_2Etoto_2Enum__dt__CASE @ A_27a @ ( c_2Etoto_2Ebit2 @ V7a ) @ V8v @ V9f @ V10f1 )
          = ( V10f1 @ V7a ) ) ) ).

thf(thm_2Etoto_2Enum__dt__size__def,axiom,
    ( ( ( c_2Etoto_2Enum__dt__size @ c_2Etoto_2Ezer )
      = c_2Enum_2E0 )
    & ! [V0a: tyop_2Etoto_2Enum__dt] :
        ( ( c_2Etoto_2Enum__dt__size @ ( c_2Etoto_2Ebit1 @ V0a ) )
        = ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Etoto_2Enum__dt__size @ V0a ) ) )
    & ! [V1a: tyop_2Etoto_2Enum__dt] :
        ( ( c_2Etoto_2Enum__dt__size @ ( c_2Etoto_2Ebit2 @ V1a ) )
        = ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Etoto_2Enum__dt__size @ V1a ) ) ) ) ).

thf(thm_2Etoto_2Enum__to__dt__primitive,axiom,
    ( c_2Etoto_2Enum__to__dt
    = ( c_2Erelation_2EWFREC @ tyop_2Enum_2Enum @ tyop_2Etoto_2Enum__dt
      @ ( c_2Emin_2E_40 @ ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o )
        @ ^ [V0R: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o] :
            ( c_2Ebool_2E_2F_5C @ ( c_2Erelation_2EWF @ tyop_2Enum_2Enum @ V0R )
            @ ( c_2Ebool_2E_2F_5C
              @ ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
                @ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1n @ c_2Enum_2E0 ) ) @ ( c_2Earithmetic_2EODD @ V1n ) ) @ ( V0R @ ( c_2Earithmetic_2EDIV2 @ ( c_2Earithmetic_2E_2D @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V1n ) ) )
              @ ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
                @ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2n @ c_2Enum_2E0 ) ) @ ( c_2Ebool_2E_7E @ ( c_2Earithmetic_2EODD @ V2n ) ) ) @ ( V0R @ ( c_2Earithmetic_2EDIV2 @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ V2n ) ) ) ) ) )
      @ ^ [V3num__to__dt: tyop_2Enum_2Enum > tyop_2Etoto_2Enum__dt,V4a: tyop_2Enum_2Enum] : ( c_2Ecombin_2EI @ tyop_2Etoto_2Enum__dt @ ( c_2Ebool_2ECOND @ tyop_2Etoto_2Enum__dt @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V4a @ c_2Enum_2E0 ) @ c_2Etoto_2Ezer @ ( c_2Ebool_2ECOND @ tyop_2Etoto_2Enum__dt @ ( c_2Earithmetic_2EODD @ V4a ) @ ( c_2Etoto_2Ebit1 @ ( V3num__to__dt @ ( c_2Earithmetic_2EDIV2 @ ( c_2Earithmetic_2E_2D @ V4a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( c_2Etoto_2Ebit2 @ ( V3num__to__dt @ ( c_2Earithmetic_2EDIV2 @ ( c_2Earithmetic_2E_2D @ V4a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ).

thf(thm_2Etoto_2Eqk__numOrd__def,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Etoto_2Eqk__numOrd @ V0m @ V1n )
      = ( c_2Etoto_2Enum__dtOrd @ ( c_2Etoto_2Enum__to__dt @ V0m ) @ ( c_2Etoto_2Enum__to__dt @ V1n ) ) ) ).

thf(thm_2Etoto_2Eqk__numto,axiom,
    ( c_2Etoto_2Eqk__numto
    = ( c_2Etoto_2ETO @ tyop_2Enum_2Enum @ c_2Etoto_2Eqk__numOrd ) ) ).

thf(thm_2Etoto_2EcharOrd,axiom,
    ! [V0a: tyop_2Estring_2Echar,V1b: tyop_2Estring_2Echar] :
      ( ( c_2Etoto_2EcharOrd @ V0a @ V1b )
      = ( c_2Etoto_2EnumOrd @ ( c_2Estring_2EORD @ V0a ) @ ( c_2Estring_2EORD @ V1b ) ) ) ).

thf(thm_2Etoto_2Echarto,axiom,
    ( c_2Etoto_2Echarto
    = ( c_2Etoto_2ETO @ tyop_2Estring_2Echar @ c_2Etoto_2EcharOrd ) ) ).

thf(thm_2Etoto_2EListOrd,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
      ( ( c_2Etoto_2EListOrd @ A_27a @ V0c )
      = ( c_2Etoto_2ETO__of__LinearOrder @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2Elistorder @ A_27a @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ) ) ).

thf(thm_2Etoto_2Elistoto,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
      ( ( c_2Etoto_2Elistoto @ A_27a @ V0c )
      = ( c_2Etoto_2ETO @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2EListOrd @ A_27a @ V0c ) ) ) ).

thf(thm_2Etoto_2Estringto,axiom,
    ( c_2Etoto_2Estringto
    = ( c_2Etoto_2Elistoto @ tyop_2Estring_2Echar @ c_2Etoto_2Echarto ) ) ).

thf(thm_2Etoto_2EimageOrd,axiom,
    ! [A_27a: $tType,A_27c: $tType,V0f: A_27a > A_27c,V1cp: A_27c > A_27c > tyop_2EternaryComparisons_2Eordering,V2a: A_27a,V3b: A_27a] :
      ( ( c_2Etoto_2EimageOrd @ A_27a @ A_27c @ V0f @ V1cp @ V2a @ V3b )
      = ( V1cp @ ( V0f @ V2a ) @ ( V0f @ V3b ) ) ) ).

thf(thm_2Etoto_2EStrongLinearOrderExists,axiom,
    ! [A_27a: $tType] :
    ? [V0R: A_27a > A_27a > $o] : ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0R ) ).

thf(thm_2Etoto_2Etrichotomous__ALT,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2Etrichotomous @ A_27a @ V0R )
    <=> ! [V1x: A_27a,V2y: A_27a] :
          ( ( ( (~) @ ( V0R @ V1x @ V2y ) )
            & ( (~) @ ( V0R @ V2y @ V1x ) ) )
         => ( V1x = V2y ) ) ) ).

thf(thm_2Etoto_2ETotOrd__TO__of__LO,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2ELinearOrder @ A_27a @ V0r )
     => ( c_2Etoto_2ETotOrd @ A_27a @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r ) ) ) ).

thf(thm_2Etoto_2ESPLIT__PAIRS,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Epair_2Eprod @ A_27a @ A_27b,V1y: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
      ( ( V0x = V1y )
    <=> ( ( ( c_2Epair_2EFST @ A_27a @ A_27b @ V0x )
          = ( c_2Epair_2EFST @ A_27a @ A_27b @ V1y ) )
        & ( ( c_2Epair_2ESND @ A_27a @ A_27b @ V0x )
          = ( c_2Epair_2ESND @ A_27a @ A_27b @ V1y ) ) ) ) ).

thf(thm_2Etoto_2Eall__cpn__distinct,axiom,
    ( ( (~) @ ( c_2EternaryComparisons_2ELESS = c_2EternaryComparisons_2EEQUAL ) )
    & ( (~) @ ( c_2EternaryComparisons_2ELESS = c_2EternaryComparisons_2EGREATER ) )
    & ( (~) @ ( c_2EternaryComparisons_2EEQUAL = c_2EternaryComparisons_2EGREATER ) )
    & ( (~) @ ( c_2EternaryComparisons_2EEQUAL = c_2EternaryComparisons_2ELESS ) )
    & ( (~) @ ( c_2EternaryComparisons_2EGREATER = c_2EternaryComparisons_2ELESS ) )
    & ( (~) @ ( c_2EternaryComparisons_2EGREATER = c_2EternaryComparisons_2EEQUAL ) ) ) ).

thf(thm_2Etoto_2ETO__exists,axiom,
    ! [A_27a: $tType] :
    ? [V0x: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] : ( c_2Etoto_2ETotOrd @ A_27a @ V0x ) ).

thf(thm_2Etoto_2ETO__apto__ID,axiom,
    ! [A_27a: $tType,V0a: tyop_2Etoto_2Etoto @ A_27a] :
      ( ( c_2Etoto_2ETO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0a ) )
      = V0a ) ).

thf(thm_2Etoto_2ETO__apto__TO__ID,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
      ( ( c_2Etoto_2ETotOrd @ A_27a @ V0r )
    <=> ( ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2ETO @ A_27a @ V0r ) )
        = V0r ) ) ).

thf(thm_2Etoto_2ETO__11,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1r_27: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
      ( ( c_2Etoto_2ETotOrd @ A_27a @ V0r )
     => ( ( c_2Etoto_2ETotOrd @ A_27a @ V1r_27 )
       => ( ( ( c_2Etoto_2ETO @ A_27a @ V0r )
            = ( c_2Etoto_2ETO @ A_27a @ V1r_27 ) )
        <=> ( V0r = V1r_27 ) ) ) ) ).

thf(thm_2Etoto_2Eonto__apto,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
      ( ( c_2Etoto_2ETotOrd @ A_27a @ V0r )
    <=> ? [V1a: tyop_2Etoto_2Etoto @ A_27a] :
          ( V0r
          = ( c_2Etoto_2Eapto @ A_27a @ V1a ) ) ) ).

thf(thm_2Etoto_2ETO__onto,axiom,
    ! [A_27a: $tType,V0a: tyop_2Etoto_2Etoto @ A_27a] :
    ? [V1r: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
      ( ( V0a
        = ( c_2Etoto_2ETO @ A_27a @ V1r ) )
      & ( c_2Etoto_2ETotOrd @ A_27a @ V1r ) ) ).

thf(thm_2Etoto_2ETotOrd__apto,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] : ( c_2Etoto_2ETotOrd @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ).

thf(thm_2Etoto_2ETO__apto__TO__IMP,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
      ( ( c_2Etoto_2ETotOrd @ A_27a @ V0r )
     => ( ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2ETO @ A_27a @ V0r ) )
        = V0r ) ) ).

thf(thm_2Etoto_2Etoto__thm,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
      ( ! [V1x: A_27a,V2y: A_27a] :
          ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
            = c_2EternaryComparisons_2EEQUAL )
        <=> ( V1x = V2y ) )
      & ! [V3x: A_27a,V4y: A_27a] :
          ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V3x @ V4y )
            = c_2EternaryComparisons_2EGREATER )
        <=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V4y @ V3x )
            = c_2EternaryComparisons_2ELESS ) )
      & ! [V5x: A_27a,V6y: A_27a,V7z: A_27a] :
          ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V5x @ V6y )
              = c_2EternaryComparisons_2ELESS )
            & ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V6y @ V7z )
              = c_2EternaryComparisons_2ELESS ) )
         => ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V5x @ V7z )
            = c_2EternaryComparisons_2ELESS ) ) ) ).

thf(thm_2Etoto_2ETO__equal__eq,axiom,
    ! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
      ( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
     => ! [V1x: A_27a,V2y: A_27a] :
          ( ( ( V0c @ V1x @ V2y )
            = c_2EternaryComparisons_2EEQUAL )
        <=> ( V1x = V2y ) ) ) ).

thf(thm_2Etoto_2Etoto__equal__eq,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
      ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
        = c_2EternaryComparisons_2EEQUAL )
    <=> ( V1x = V2y ) ) ).

thf(thm_2Etoto_2Etoto__equal__imp__eq,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
      ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
        = c_2EternaryComparisons_2EEQUAL )
     => ( V1x = V2y ) ) ).

thf(thm_2Etoto_2ETO__refl,axiom,
    ! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
      ( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
     => ! [V1x: A_27a] :
          ( ( V0c @ V1x @ V1x )
          = c_2EternaryComparisons_2EEQUAL ) ) ).

thf(thm_2Etoto_2Etoto__refl,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a] :
      ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V1x )
      = c_2EternaryComparisons_2EEQUAL ) ).

thf(thm_2Etoto_2Etoto__equal__sym,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
      ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
        = c_2EternaryComparisons_2EEQUAL )
    <=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V1x )
        = c_2EternaryComparisons_2EEQUAL ) ) ).

thf(thm_2Etoto_2ETO__antisym,axiom,
    ! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
      ( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
     => ! [V1x: A_27a,V2y: A_27a] :
          ( ( ( V0c @ V1x @ V2y )
            = c_2EternaryComparisons_2EGREATER )
        <=> ( ( V0c @ V2y @ V1x )
            = c_2EternaryComparisons_2ELESS ) ) ) ).

thf(thm_2Etoto_2Etoto__antisym,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
      ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
        = c_2EternaryComparisons_2EGREATER )
    <=> ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V1x )
        = c_2EternaryComparisons_2ELESS ) ) ).

thf(thm_2Etoto_2Etoto__not__less__refl,axiom,
    ! [A_27a: $tType,V0cmp: tyop_2Etoto_2Etoto @ A_27a,V1h: A_27a] :
      ( ( ( c_2Etoto_2Eapto @ A_27a @ V0cmp @ V1h @ V1h )
        = c_2EternaryComparisons_2ELESS )
    <=> c_2Ebool_2EF ) ).

thf(thm_2Etoto_2Etoto__swap__cases,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
      ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V1x )
      = ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y ) @ c_2EternaryComparisons_2EGREATER @ c_2EternaryComparisons_2EEQUAL @ c_2EternaryComparisons_2ELESS ) ) ).

thf(thm_2Etoto_2Etoto__glneq,axiom,
    ! [A_27a: $tType] :
      ( ! [V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
          ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
            = c_2EternaryComparisons_2ELESS )
         => ( (~) @ ( V1x = V2y ) ) )
      & ! [V3c: tyop_2Etoto_2Etoto @ A_27a,V4x: A_27a,V5y: A_27a] :
          ( ( ( c_2Etoto_2Eapto @ A_27a @ V3c @ V4x @ V5y )
            = c_2EternaryComparisons_2EGREATER )
         => ( (~) @ ( V4x = V5y ) ) ) ) ).

thf(thm_2Etoto_2Etoto__cpn__eqn,axiom,
    ! [A_27a: $tType] :
      ( ! [V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
          ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
            = c_2EternaryComparisons_2EEQUAL )
         => ( V1x = V2y ) )
      & ! [V3c: tyop_2Etoto_2Etoto @ A_27a,V4x: A_27a,V5y: A_27a] :
          ( ( ( c_2Etoto_2Eapto @ A_27a @ V3c @ V4x @ V5y )
            = c_2EternaryComparisons_2ELESS )
         => ( (~) @ ( V4x = V5y ) ) )
      & ! [V6c: tyop_2Etoto_2Etoto @ A_27a,V7x: A_27a,V8y: A_27a] :
          ( ( ( c_2Etoto_2Eapto @ A_27a @ V6c @ V7x @ V8y )
            = c_2EternaryComparisons_2EGREATER )
         => ( (~) @ ( V7x = V8y ) ) ) ) ).

thf(thm_2Etoto_2ETO__cpn__eqn,axiom,
    ! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
      ( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
     => ( ! [V1x: A_27a,V2y: A_27a] :
            ( ( ( V0c @ V1x @ V2y )
              = c_2EternaryComparisons_2ELESS )
           => ( (~) @ ( V1x = V2y ) ) )
        & ! [V3x: A_27a,V4y: A_27a] :
            ( ( ( V0c @ V3x @ V4y )
              = c_2EternaryComparisons_2EGREATER )
           => ( (~) @ ( V3x = V4y ) ) )
        & ! [V5x: A_27a,V6y: A_27a] :
            ( ( ( V0c @ V5x @ V6y )
              = c_2EternaryComparisons_2EEQUAL )
           => ( V5x = V6y ) ) ) ) ).

thf(thm_2Etoto_2ENOT__EQ__LESS__IMP,axiom,
    ! [A_27a: $tType,V0cmp: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a] :
      ( ( (~)
        @ ( ( c_2Etoto_2Eapto @ A_27a @ V0cmp @ V1x @ V2y )
          = c_2EternaryComparisons_2ELESS ) )
     => ( ( V1x = V2y )
        | ( ( c_2Etoto_2Eapto @ A_27a @ V0cmp @ V2y @ V1x )
          = c_2EternaryComparisons_2ELESS ) ) ) ).

thf(thm_2Etoto_2EtotoEEtrans,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
      ( ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
            = c_2EternaryComparisons_2EEQUAL )
          & ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V3z )
            = c_2EternaryComparisons_2EEQUAL ) )
       => ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
          = c_2EternaryComparisons_2EEQUAL ) )
      & ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
            = c_2EternaryComparisons_2EEQUAL )
          & ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V3z @ V2y )
            = c_2EternaryComparisons_2EEQUAL ) )
       => ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
          = c_2EternaryComparisons_2EEQUAL ) ) ) ).

thf(thm_2Etoto_2EtotoLLtrans,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
      ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
          = c_2EternaryComparisons_2ELESS )
        & ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V3z )
          = c_2EternaryComparisons_2ELESS ) )
     => ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
        = c_2EternaryComparisons_2ELESS ) ) ).

thf(thm_2Etoto_2EtotoLGtrans,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
      ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
          = c_2EternaryComparisons_2ELESS )
        & ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V3z @ V2y )
          = c_2EternaryComparisons_2EGREATER ) )
     => ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
        = c_2EternaryComparisons_2ELESS ) ) ).

thf(thm_2Etoto_2EtotoGGtrans,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
      ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V1x )
          = c_2EternaryComparisons_2EGREATER )
        & ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V3z @ V2y )
          = c_2EternaryComparisons_2EGREATER ) )
     => ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
        = c_2EternaryComparisons_2ELESS ) ) ).

thf(thm_2Etoto_2EtotoGLtrans,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
      ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V1x )
          = c_2EternaryComparisons_2EGREATER )
        & ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V3z )
          = c_2EternaryComparisons_2ELESS ) )
     => ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
        = c_2EternaryComparisons_2ELESS ) ) ).

thf(thm_2Etoto_2EtotoLEtrans,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
      ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
          = c_2EternaryComparisons_2ELESS )
        & ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V3z )
          = c_2EternaryComparisons_2EEQUAL ) )
     => ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
        = c_2EternaryComparisons_2ELESS ) ) ).

thf(thm_2Etoto_2EtotoELtrans,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
      ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
          = c_2EternaryComparisons_2EEQUAL )
        & ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V3z )
          = c_2EternaryComparisons_2ELESS ) )
     => ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
        = c_2EternaryComparisons_2ELESS ) ) ).

thf(thm_2Etoto_2Etoto__trans__less,axiom,
    ! [A_27a: $tType] :
      ( ! [V0c: tyop_2Etoto_2Etoto @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
          ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V2y )
              = c_2EternaryComparisons_2ELESS )
            & ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2y @ V3z )
              = c_2EternaryComparisons_2ELESS ) )
         => ( ( c_2Etoto_2Eapto @ A_27a @ V0c @ V1x @ V3z )
            = c_2EternaryComparisons_2ELESS ) )
      & ! [V4c: tyop_2Etoto_2Etoto @ A_27a,V5x: A_27a,V6y: A_27a,V7z: A_27a] :
          ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V4c @ V5x @ V6y )
              = c_2EternaryComparisons_2ELESS )
            & ( ( c_2Etoto_2Eapto @ A_27a @ V4c @ V7z @ V6y )
              = c_2EternaryComparisons_2EGREATER ) )
         => ( ( c_2Etoto_2Eapto @ A_27a @ V4c @ V5x @ V7z )
            = c_2EternaryComparisons_2ELESS ) )
      & ! [V8c: tyop_2Etoto_2Etoto @ A_27a,V9x: A_27a,V10y: A_27a,V11z: A_27a] :
          ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V8c @ V10y @ V9x )
              = c_2EternaryComparisons_2EGREATER )
            & ( ( c_2Etoto_2Eapto @ A_27a @ V8c @ V11z @ V10y )
              = c_2EternaryComparisons_2EGREATER ) )
         => ( ( c_2Etoto_2Eapto @ A_27a @ V8c @ V9x @ V11z )
            = c_2EternaryComparisons_2ELESS ) )
      & ! [V12c: tyop_2Etoto_2Etoto @ A_27a,V13x: A_27a,V14y: A_27a,V15z: A_27a] :
          ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V12c @ V14y @ V13x )
              = c_2EternaryComparisons_2EGREATER )
            & ( ( c_2Etoto_2Eapto @ A_27a @ V12c @ V14y @ V15z )
              = c_2EternaryComparisons_2ELESS ) )
         => ( ( c_2Etoto_2Eapto @ A_27a @ V12c @ V13x @ V15z )
            = c_2EternaryComparisons_2ELESS ) )
      & ! [V16c: tyop_2Etoto_2Etoto @ A_27a,V17x: A_27a,V18y: A_27a,V19z: A_27a] :
          ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V16c @ V17x @ V18y )
              = c_2EternaryComparisons_2ELESS )
            & ( ( c_2Etoto_2Eapto @ A_27a @ V16c @ V18y @ V19z )
              = c_2EternaryComparisons_2EEQUAL ) )
         => ( ( c_2Etoto_2Eapto @ A_27a @ V16c @ V17x @ V19z )
            = c_2EternaryComparisons_2ELESS ) )
      & ! [V20c: tyop_2Etoto_2Etoto @ A_27a,V21x: A_27a,V22y: A_27a,V23z: A_27a] :
          ( ( ( ( c_2Etoto_2Eapto @ A_27a @ V20c @ V21x @ V22y )
              = c_2EternaryComparisons_2EEQUAL )
            & ( ( c_2Etoto_2Eapto @ A_27a @ V20c @ V22y @ V23z )
              = c_2EternaryComparisons_2ELESS ) )
         => ( ( c_2Etoto_2Eapto @ A_27a @ V20c @ V21x @ V23z )
            = c_2EternaryComparisons_2ELESS ) ) ) ).

thf(thm_2Etoto_2EWeak__Weak__of,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] : ( c_2Erelation_2EWeakLinearOrder @ A_27a @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ).

thf(thm_2Etoto_2ESTRORD__SLO,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2EWeakLinearOrder @ A_27a @ V0R )
     => ( c_2Erelation_2EStrongLinearOrder @ A_27a @ ( c_2Erelation_2ESTRORD @ A_27a @ V0R ) ) ) ).

thf(thm_2Etoto_2EStrongof__toto__STRORD,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
      ( ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) )
      = ( c_2Erelation_2ESTRORD @ A_27a @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ) ).

thf(thm_2Etoto_2EStrong__Strong__of,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] : ( c_2Erelation_2EStrongLinearOrder @ A_27a @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ).

thf(thm_2Etoto_2EStrong__Strong__of__TO,axiom,
    ! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
      ( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
     => ( c_2Erelation_2EStrongLinearOrder @ A_27a @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ V0c ) ) ) ).

thf(thm_2Etoto_2ETotOrd__TO__of__Weak,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2EWeakLinearOrder @ A_27a @ V0r )
     => ( c_2Etoto_2ETotOrd @ A_27a @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r ) ) ) ).

thf(thm_2Etoto_2ETotOrd__TO__of__Strong,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0r )
     => ( c_2Etoto_2ETotOrd @ A_27a @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r ) ) ) ).

thf(thm_2Etoto_2Etoto__Weak__thm,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
      ( ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) )
      = V0c ) ).

thf(thm_2Etoto_2Etoto__Strong__thm,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
      ( ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) )
      = V0c ) ).

thf(thm_2Etoto_2EWeak__toto__thm,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2EWeakLinearOrder @ A_27a @ V0r )
     => ( ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a @ V0r ) ) )
        = V0r ) ) ).

thf(thm_2Etoto_2EStrong__toto__thm,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0r )
     => ( ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a @ V0r ) ) )
        = V0r ) ) ).

thf(thm_2Etoto_2ETotOrd__inv,axiom,
    ! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
      ( ( c_2Etoto_2ETotOrd @ A_27a @ V0c )
     => ( c_2Etoto_2ETotOrd @ A_27a @ ( c_2Etoto_2ETO__inv @ A_27a @ V0c ) ) ) ).

thf(thm_2Etoto_2Einv__TO,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
      ( ( c_2Etoto_2ETotOrd @ A_27a @ V0r )
     => ( ( c_2Etoto_2Etoto__inv @ A_27a @ ( c_2Etoto_2ETO @ A_27a @ V0r ) )
        = ( c_2Etoto_2ETO @ A_27a @ ( c_2Etoto_2ETO__inv @ A_27a @ V0r ) ) ) ) ).

thf(thm_2Etoto_2Eapto__inv,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
      ( ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2Etoto__inv @ A_27a @ V0c ) )
      = ( c_2Etoto_2ETO__inv @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ).

thf(thm_2Etoto_2EWeak__toto__inv,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
      ( ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2Etoto__inv @ A_27a @ V0c ) ) )
      = ( c_2Erelation_2Einv @ A_27a @ A_27a @ ( c_2Etoto_2EWeakLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ) ).

thf(thm_2Etoto_2EStrong__toto__inv,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
      ( ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ ( c_2Etoto_2Etoto__inv @ A_27a @ V0c ) ) )
      = ( c_2Erelation_2Einv @ A_27a @ A_27a @ ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2Eapto @ A_27a @ V0c ) ) ) ) ).

thf(thm_2Etoto_2ETO__inv__TO__inv,axiom,
    ! [A_27a: $tType,V0c: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering] :
      ( ( c_2Etoto_2ETO__inv @ A_27a @ ( c_2Etoto_2ETO__inv @ A_27a @ V0c ) )
      = V0c ) ).

thf(thm_2Etoto_2Etoto__inv__toto__inv,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
      ( ( c_2Etoto_2Etoto__inv @ A_27a @ ( c_2Etoto_2Etoto__inv @ A_27a @ V0c ) )
      = V0c ) ).

thf(thm_2Etoto_2ETO__inv__Ord,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
      ( ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0r ) )
      = ( c_2Etoto_2ETO__inv @ A_27a @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r ) ) ) ).

thf(thm_2Etoto_2ETO__of__less__rel,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0r )
     => ! [V1x: A_27a,V2y: A_27a] :
          ( ( ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r @ V1x @ V2y )
            = c_2EternaryComparisons_2ELESS )
        <=> ( V0r @ V1x @ V2y ) ) ) ).

thf(thm_2Etoto_2ETO__of__greater__ler,axiom,
    ! [A_27a: $tType,V0r: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0r )
     => ! [V1x: A_27a,V2y: A_27a] :
          ( ( ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0r @ V1x @ V2y )
            = c_2EternaryComparisons_2EGREATER )
        <=> ( V0r @ V2y @ V1x ) ) ) ).

thf(thm_2Etoto_2Etoto__equal__imp,axiom,
    ! [A_27a: $tType,V0cmp: tyop_2Etoto_2Etoto @ A_27a,V1phi: A_27a > A_27a > $o] :
      ( ( ( c_2Erelation_2ELinearOrder @ A_27a @ V1phi )
        & ( V0cmp
          = ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a @ V1phi ) ) )
     => ! [V2x: A_27a,V3y: A_27a] :
          ( ( ( V2x = V3y )
          <=> c_2Ebool_2ET )
         => ( ( c_2Etoto_2Eapto @ A_27a @ V0cmp @ V2x @ V3y )
            = c_2EternaryComparisons_2EEQUAL ) ) ) ).

thf(thm_2Etoto_2Etoto__unequal__imp,axiom,
    ! [A_27a: $tType,V0cmp: tyop_2Etoto_2Etoto @ A_27a,V1phi: A_27a > A_27a > $o] :
      ( ( ( c_2Erelation_2ELinearOrder @ A_27a @ V1phi )
        & ( V0cmp
          = ( c_2Etoto_2Etoto__of__LinearOrder @ A_27a @ V1phi ) ) )
     => ! [V2x: A_27a,V3y: A_27a] :
          ( ( ( V2x = V3y )
          <=> c_2Ebool_2EF )
         => ( c_2Ebool_2ECOND @ $o @ ( V1phi @ V2x @ V3y ) @ ( c_2Emin_2E_3D @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2Eapto @ A_27a @ V0cmp @ V2x @ V3y ) @ c_2EternaryComparisons_2ELESS ) @ ( c_2Emin_2E_3D @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2Eapto @ A_27a @ V0cmp @ V2x @ V3y ) @ c_2EternaryComparisons_2EGREATER ) ) ) ) ).

thf(thm_2Etoto_2EStrongOrder__ALT,axiom,
    ! [A_27a: $tType,V0Z: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2EStrongOrder @ A_27a @ V0Z )
    <=> ( ( c_2Erelation_2Eirreflexive @ A_27a @ V0Z )
        & ( c_2Erelation_2Etransitive @ A_27a @ V0Z ) ) ) ).

thf(thm_2Etoto_2ELEX__ALT,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1U: A_27b > A_27b > $o,V2c: tyop_2Epair_2Eprod @ A_27a @ A_27b,V3d: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
      ( ( c_2Epair_2ELEX @ A_27a @ A_27b @ V0R @ V1U @ V2c @ V3d )
    <=> ( ( V0R @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V2c ) @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V3d ) )
        | ( ( ( c_2Epair_2EFST @ A_27a @ A_27b @ V2c )
            = ( c_2Epair_2EFST @ A_27a @ A_27b @ V3d ) )
          & ( V1U @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V2c ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V3d ) ) ) ) ) ).

thf(thm_2Etoto_2ESLO__LEX,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1V: A_27b > A_27b > $o] :
      ( ( ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0R )
        & ( c_2Erelation_2EStrongLinearOrder @ A_27b @ V1V ) )
     => ( c_2Erelation_2EStrongLinearOrder @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2ELEX @ A_27a @ A_27b @ V0R @ V1V ) ) ) ).

thf(thm_2Etoto_2ElexTO__thm,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1V: A_27b > A_27b > tyop_2EternaryComparisons_2Eordering] :
      ( ( ( c_2Etoto_2ETotOrd @ A_27a @ V0R )
        & ( c_2Etoto_2ETotOrd @ A_27b @ V1V ) )
     => ! [V2x: tyop_2Epair_2Eprod @ A_27a @ A_27b,V3y: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
          ( ( c_2Etoto_2ElexTO @ A_27a @ A_27b @ V0R @ V1V @ V2x @ V3y )
          = ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( V0R @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V2x ) @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V3y ) ) @ c_2EternaryComparisons_2ELESS @ ( V1V @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V2x ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V3y ) ) @ c_2EternaryComparisons_2EGREATER ) ) ) ).

thf(thm_2Etoto_2ElexTO__ALT,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1V: A_27b > A_27b > tyop_2EternaryComparisons_2Eordering] :
      ( ( ( c_2Etoto_2ETotOrd @ A_27a @ V0R )
        & ( c_2Etoto_2ETotOrd @ A_27b @ V1V ) )
     => ( c_2Ebool_2E_21 @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
        @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o
          @ ^ [V2r: A_27a,V3u: A_27b] :
              ( c_2Ebool_2E_21 @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
              @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o
                @ ^ [V4r_27: A_27a,V5u_27: A_27b] : ( c_2Emin_2E_3D @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2ElexTO @ A_27a @ A_27b @ V0R @ V1V @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2r @ V3u ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V4r_27 @ V5u_27 ) ) @ ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( V0R @ V2r @ V4r_27 ) @ c_2EternaryComparisons_2ELESS @ ( V1V @ V3u @ V5u_27 ) @ c_2EternaryComparisons_2EGREATER ) ) ) ) ) ) ) ).

thf(thm_2Etoto_2ETO__lexTO,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > tyop_2EternaryComparisons_2Eordering,V1V: A_27b > A_27b > tyop_2EternaryComparisons_2Eordering] :
      ( ( ( c_2Etoto_2ETotOrd @ A_27a @ V0R )
        & ( c_2Etoto_2ETotOrd @ A_27b @ V1V ) )
     => ( c_2Etoto_2ETotOrd @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Etoto_2ElexTO @ A_27a @ A_27b @ V0R @ V1V ) ) ) ).

thf(thm_2Etoto_2Epre__aplextoto,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1v: tyop_2Etoto_2Etoto @ A_27b,V2x: tyop_2Epair_2Eprod @ A_27a @ A_27b,V3y: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
      ( ( c_2Etoto_2Eapto @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Etoto_2Elextoto @ A_27a @ A_27b @ V0c @ V1v ) @ V2x @ V3y )
      = ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2Eapto @ A_27a @ V0c @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V2x ) @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V3y ) ) @ c_2EternaryComparisons_2ELESS @ ( c_2Etoto_2Eapto @ A_27b @ V1v @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V2x ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V3y ) ) @ c_2EternaryComparisons_2EGREATER ) ) ).

thf(thm_2Etoto_2Eaplextoto,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a,V1v: tyop_2Etoto_2Etoto @ A_27b,V2x1: A_27a,V3x2: A_27b,V4y1: A_27a,V5y2: A_27b] :
      ( ( c_2Etoto_2Eapto @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Etoto_2Elextoto @ A_27a @ A_27b @ V0c @ V1v ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x1 @ V3x2 ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V4y1 @ V5y2 ) )
      = ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2Eapto @ A_27a @ V0c @ V2x1 @ V4y1 ) @ c_2EternaryComparisons_2ELESS @ ( c_2Etoto_2Eapto @ A_27b @ V1v @ V3x2 @ V5y2 ) @ c_2EternaryComparisons_2EGREATER ) ) ).

thf(thm_2Etoto_2EStrongLinearOrder__LESS,axiom,
    c_2Erelation_2EStrongLinearOrder @ tyop_2Enum_2Enum @ c_2Eprim__rec_2E_3C ).

thf(thm_2Etoto_2ETO__numOrd,axiom,
    c_2Etoto_2ETotOrd @ tyop_2Enum_2Enum @ c_2Etoto_2EnumOrd ).

thf(thm_2Etoto_2Eapnumto__thm,axiom,
    ( ( c_2Etoto_2Eapto @ tyop_2Enum_2Enum @ c_2Etoto_2Enumto )
    = c_2Etoto_2EnumOrd ) ).

thf(thm_2Etoto_2EnumeralOrd,axiom,
    ! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] :
      ( ( ( c_2Etoto_2EnumOrd @ c_2Earithmetic_2EZERO @ c_2Earithmetic_2EZERO )
        = c_2EternaryComparisons_2EEQUAL )
      & ( ( c_2Etoto_2EnumOrd @ c_2Earithmetic_2EZERO @ ( c_2Earithmetic_2EBIT1 @ V1y ) )
        = c_2EternaryComparisons_2ELESS )
      & ( ( c_2Etoto_2EnumOrd @ c_2Earithmetic_2EZERO @ ( c_2Earithmetic_2EBIT2 @ V1y ) )
        = c_2EternaryComparisons_2ELESS )
      & ( ( c_2Etoto_2EnumOrd @ ( c_2Earithmetic_2EBIT1 @ V0x ) @ c_2Earithmetic_2EZERO )
        = c_2EternaryComparisons_2EGREATER )
      & ( ( c_2Etoto_2EnumOrd @ ( c_2Earithmetic_2EBIT2 @ V0x ) @ c_2Earithmetic_2EZERO )
        = c_2EternaryComparisons_2EGREATER )
      & ( ( c_2Etoto_2EnumOrd @ ( c_2Earithmetic_2EBIT1 @ V0x ) @ ( c_2Earithmetic_2EBIT1 @ V1y ) )
        = ( c_2Etoto_2EnumOrd @ V0x @ V1y ) )
      & ( ( c_2Etoto_2EnumOrd @ ( c_2Earithmetic_2EBIT2 @ V0x ) @ ( c_2Earithmetic_2EBIT2 @ V1y ) )
        = ( c_2Etoto_2EnumOrd @ V0x @ V1y ) )
      & ( ( c_2Etoto_2EnumOrd @ ( c_2Earithmetic_2EBIT1 @ V0x ) @ ( c_2Earithmetic_2EBIT2 @ V1y ) )
        = ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2EnumOrd @ V0x @ V1y ) @ c_2EternaryComparisons_2ELESS @ c_2EternaryComparisons_2ELESS @ c_2EternaryComparisons_2EGREATER ) )
      & ( ( c_2Etoto_2EnumOrd @ ( c_2Earithmetic_2EBIT2 @ V0x ) @ ( c_2Earithmetic_2EBIT1 @ V1y ) )
        = ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2EnumOrd @ V0x @ V1y ) @ c_2EternaryComparisons_2ELESS @ c_2EternaryComparisons_2EGREATER @ c_2EternaryComparisons_2EGREATER ) ) ) ).

thf(thm_2Etoto_2Edatatype__num__dt,axiom,
    ! [V0num__dt: tyop_2Etoto_2Enum__dt > ( tyop_2Etoto_2Enum__dt > tyop_2Etoto_2Enum__dt ) > ( tyop_2Etoto_2Enum__dt > tyop_2Etoto_2Enum__dt ) > $o] : ( c_2Ebool_2EDATATYPE @ $o @ ( V0num__dt @ c_2Etoto_2Ezer @ c_2Etoto_2Ebit1 @ c_2Etoto_2Ebit2 ) ) ).

thf(thm_2Etoto_2Enum__dt__11,axiom,
    ( ! [V0a: tyop_2Etoto_2Enum__dt,V1a_27: tyop_2Etoto_2Enum__dt] :
        ( ( ( c_2Etoto_2Ebit1 @ V0a )
          = ( c_2Etoto_2Ebit1 @ V1a_27 ) )
      <=> ( V0a = V1a_27 ) )
    & ! [V2a: tyop_2Etoto_2Enum__dt,V3a_27: tyop_2Etoto_2Enum__dt] :
        ( ( ( c_2Etoto_2Ebit2 @ V2a )
          = ( c_2Etoto_2Ebit2 @ V3a_27 ) )
      <=> ( V2a = V3a_27 ) ) ) ).

thf(thm_2Etoto_2Enum__dt__distinct,axiom,
    ( ! [V0a: tyop_2Etoto_2Enum__dt] :
        ( (~)
        @ ( c_2Etoto_2Ezer
          = ( c_2Etoto_2Ebit1 @ V0a ) ) )
    & ! [V1a: tyop_2Etoto_2Enum__dt] :
        ( (~)
        @ ( c_2Etoto_2Ezer
          = ( c_2Etoto_2Ebit2 @ V1a ) ) )
    & ! [V2a_27: tyop_2Etoto_2Enum__dt,V3a: tyop_2Etoto_2Enum__dt] :
        ( (~)
        @ ( ( c_2Etoto_2Ebit1 @ V3a )
          = ( c_2Etoto_2Ebit2 @ V2a_27 ) ) ) ) ).

thf(thm_2Etoto_2Enum__dt__nchotomy,axiom,
    ! [V0nn: tyop_2Etoto_2Enum__dt] :
      ( ( V0nn = c_2Etoto_2Ezer )
      | ? [V1n: tyop_2Etoto_2Enum__dt] :
          ( V0nn
          = ( c_2Etoto_2Ebit1 @ V1n ) )
      | ? [V2n: tyop_2Etoto_2Enum__dt] :
          ( V0nn
          = ( c_2Etoto_2Ebit2 @ V2n ) ) ) ).

thf(thm_2Etoto_2Enum__dt__Axiom,axiom,
    ! [A_27a: $tType,V0f0: A_27a,V1f1: tyop_2Etoto_2Enum__dt > A_27a > A_27a,V2f2: tyop_2Etoto_2Enum__dt > A_27a > A_27a] :
    ? [V3fn: tyop_2Etoto_2Enum__dt > A_27a] :
      ( ( ( V3fn @ c_2Etoto_2Ezer )
        = V0f0 )
      & ! [V4a: tyop_2Etoto_2Enum__dt] :
          ( ( V3fn @ ( c_2Etoto_2Ebit1 @ V4a ) )
          = ( V1f1 @ V4a @ ( V3fn @ V4a ) ) )
      & ! [V5a: tyop_2Etoto_2Enum__dt] :
          ( ( V3fn @ ( c_2Etoto_2Ebit2 @ V5a ) )
          = ( V2f2 @ V5a @ ( V3fn @ V5a ) ) ) ) ).

thf(thm_2Etoto_2Enum__dt__induction,axiom,
    ! [V0P: tyop_2Etoto_2Enum__dt > $o] :
      ( ( ( V0P @ c_2Etoto_2Ezer )
        & ! [V1n: tyop_2Etoto_2Enum__dt] :
            ( ( V0P @ V1n )
           => ( V0P @ ( c_2Etoto_2Ebit1 @ V1n ) ) )
        & ! [V2n: tyop_2Etoto_2Enum__dt] :
            ( ( V0P @ V2n )
           => ( V0P @ ( c_2Etoto_2Ebit2 @ V2n ) ) ) )
     => ! [V3n: tyop_2Etoto_2Enum__dt] : ( V0P @ V3n ) ) ).

thf(thm_2Etoto_2Enum__dt__case__cong,axiom,
    ! [A_27a: $tType,V0v_27: A_27a,V1f1_27: tyop_2Etoto_2Enum__dt > A_27a,V2f_27: tyop_2Etoto_2Enum__dt > A_27a,V3M: tyop_2Etoto_2Enum__dt,V4M_27: tyop_2Etoto_2Enum__dt,V5v: A_27a,V6f: tyop_2Etoto_2Enum__dt > A_27a,V7f1: tyop_2Etoto_2Enum__dt > A_27a] :
      ( ( ( V3M = V4M_27 )
        & ( ( V4M_27 = c_2Etoto_2Ezer )
         => ( V5v = V0v_27 ) )
        & ! [V8a: tyop_2Etoto_2Enum__dt] :
            ( ( V4M_27
              = ( c_2Etoto_2Ebit1 @ V8a ) )
           => ( ( V6f @ V8a )
              = ( V2f_27 @ V8a ) ) )
        & ! [V9a: tyop_2Etoto_2Enum__dt] :
            ( ( V4M_27
              = ( c_2Etoto_2Ebit2 @ V9a ) )
           => ( ( V7f1 @ V9a )
              = ( V1f1_27 @ V9a ) ) ) )
     => ( ( c_2Etoto_2Enum__dt__CASE @ A_27a @ V3M @ V5v @ V6f @ V7f1 )
        = ( c_2Etoto_2Enum__dt__CASE @ A_27a @ V4M_27 @ V0v_27 @ V2f_27 @ V1f1_27 ) ) ) ).

thf(thm_2Etoto_2Enum__dt__case__eq,axiom,
    ! [A_27a: $tType,V0x: tyop_2Etoto_2Enum__dt,V1v_27: A_27a,V2v: A_27a,V3f1: tyop_2Etoto_2Enum__dt > A_27a,V4f: tyop_2Etoto_2Enum__dt > A_27a] :
      ( ( ( c_2Etoto_2Enum__dt__CASE @ A_27a @ V0x @ V2v @ V4f @ V3f1 )
        = V1v_27 )
    <=> ( ( ( V0x = c_2Etoto_2Ezer )
          & ( V2v = V1v_27 ) )
        | ? [V5n: tyop_2Etoto_2Enum__dt] :
            ( ( V0x
              = ( c_2Etoto_2Ebit1 @ V5n ) )
            & ( ( V4f @ V5n )
              = V1v_27 ) )
        | ? [V6n: tyop_2Etoto_2Enum__dt] :
            ( ( V0x
              = ( c_2Etoto_2Ebit2 @ V6n ) )
            & ( ( V3f1 @ V6n )
              = V1v_27 ) ) ) ) ).

thf(thm_2Etoto_2Enum__dtOrd__ind,axiom,
    ! [V0P: tyop_2Etoto_2Enum__dt > tyop_2Etoto_2Enum__dt > $o] :
      ( ( ( V0P @ c_2Etoto_2Ezer @ c_2Etoto_2Ezer )
        & ! [V1x: tyop_2Etoto_2Enum__dt] : ( V0P @ c_2Etoto_2Ezer @ ( c_2Etoto_2Ebit1 @ V1x ) )
        & ! [V2x: tyop_2Etoto_2Enum__dt] : ( V0P @ c_2Etoto_2Ezer @ ( c_2Etoto_2Ebit2 @ V2x ) )
        & ! [V3x: tyop_2Etoto_2Enum__dt] : ( V0P @ ( c_2Etoto_2Ebit1 @ V3x ) @ c_2Etoto_2Ezer )
        & ! [V4x: tyop_2Etoto_2Enum__dt] : ( V0P @ ( c_2Etoto_2Ebit2 @ V4x ) @ c_2Etoto_2Ezer )
        & ! [V5x: tyop_2Etoto_2Enum__dt,V6y: tyop_2Etoto_2Enum__dt] : ( V0P @ ( c_2Etoto_2Ebit1 @ V5x ) @ ( c_2Etoto_2Ebit2 @ V6y ) )
        & ! [V7x: tyop_2Etoto_2Enum__dt,V8y: tyop_2Etoto_2Enum__dt] : ( V0P @ ( c_2Etoto_2Ebit2 @ V7x ) @ ( c_2Etoto_2Ebit1 @ V8y ) )
        & ! [V9x: tyop_2Etoto_2Enum__dt,V10y: tyop_2Etoto_2Enum__dt] :
            ( ( V0P @ V9x @ V10y )
           => ( V0P @ ( c_2Etoto_2Ebit1 @ V9x ) @ ( c_2Etoto_2Ebit1 @ V10y ) ) )
        & ! [V11x: tyop_2Etoto_2Enum__dt,V12y: tyop_2Etoto_2Enum__dt] :
            ( ( V0P @ V11x @ V12y )
           => ( V0P @ ( c_2Etoto_2Ebit2 @ V11x ) @ ( c_2Etoto_2Ebit2 @ V12y ) ) ) )
     => ! [V13v: tyop_2Etoto_2Enum__dt,V14v1: tyop_2Etoto_2Enum__dt] : ( V0P @ V13v @ V14v1 ) ) ).

thf(thm_2Etoto_2Enum__dtOrd,axiom,
    ( ( ( c_2Etoto_2Enum__dtOrd @ c_2Etoto_2Ezer @ c_2Etoto_2Ezer )
      = c_2EternaryComparisons_2EEQUAL )
    & ! [V0x: tyop_2Etoto_2Enum__dt] :
        ( ( c_2Etoto_2Enum__dtOrd @ c_2Etoto_2Ezer @ ( c_2Etoto_2Ebit1 @ V0x ) )
        = c_2EternaryComparisons_2ELESS )
    & ! [V1x: tyop_2Etoto_2Enum__dt] :
        ( ( c_2Etoto_2Enum__dtOrd @ c_2Etoto_2Ezer @ ( c_2Etoto_2Ebit2 @ V1x ) )
        = c_2EternaryComparisons_2ELESS )
    & ! [V2x: tyop_2Etoto_2Enum__dt] :
        ( ( c_2Etoto_2Enum__dtOrd @ ( c_2Etoto_2Ebit1 @ V2x ) @ c_2Etoto_2Ezer )
        = c_2EternaryComparisons_2EGREATER )
    & ! [V3x: tyop_2Etoto_2Enum__dt] :
        ( ( c_2Etoto_2Enum__dtOrd @ ( c_2Etoto_2Ebit2 @ V3x ) @ c_2Etoto_2Ezer )
        = c_2EternaryComparisons_2EGREATER )
    & ! [V4y: tyop_2Etoto_2Enum__dt,V5x: tyop_2Etoto_2Enum__dt] :
        ( ( c_2Etoto_2Enum__dtOrd @ ( c_2Etoto_2Ebit1 @ V5x ) @ ( c_2Etoto_2Ebit2 @ V4y ) )
        = c_2EternaryComparisons_2ELESS )
    & ! [V6y: tyop_2Etoto_2Enum__dt,V7x: tyop_2Etoto_2Enum__dt] :
        ( ( c_2Etoto_2Enum__dtOrd @ ( c_2Etoto_2Ebit2 @ V7x ) @ ( c_2Etoto_2Ebit1 @ V6y ) )
        = c_2EternaryComparisons_2EGREATER )
    & ! [V8y: tyop_2Etoto_2Enum__dt,V9x: tyop_2Etoto_2Enum__dt] :
        ( ( c_2Etoto_2Enum__dtOrd @ ( c_2Etoto_2Ebit1 @ V9x ) @ ( c_2Etoto_2Ebit1 @ V8y ) )
        = ( c_2Etoto_2Enum__dtOrd @ V9x @ V8y ) )
    & ! [V10y: tyop_2Etoto_2Enum__dt,V11x: tyop_2Etoto_2Enum__dt] :
        ( ( c_2Etoto_2Enum__dtOrd @ ( c_2Etoto_2Ebit2 @ V11x ) @ ( c_2Etoto_2Ebit2 @ V10y ) )
        = ( c_2Etoto_2Enum__dtOrd @ V11x @ V10y ) ) ) ).

thf(thm_2Etoto_2ETO__qk__numOrd,axiom,
    c_2Etoto_2ETotOrd @ tyop_2Enum_2Enum @ c_2Etoto_2Eqk__numOrd ).

thf(thm_2Etoto_2Eqk__numeralOrd,axiom,
    ! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] :
      ( ( ( c_2Etoto_2Eqk__numOrd @ c_2Earithmetic_2EZERO @ c_2Earithmetic_2EZERO )
        = c_2EternaryComparisons_2EEQUAL )
      & ( ( c_2Etoto_2Eqk__numOrd @ c_2Earithmetic_2EZERO @ ( c_2Earithmetic_2EBIT1 @ V1y ) )
        = c_2EternaryComparisons_2ELESS )
      & ( ( c_2Etoto_2Eqk__numOrd @ c_2Earithmetic_2EZERO @ ( c_2Earithmetic_2EBIT2 @ V1y ) )
        = c_2EternaryComparisons_2ELESS )
      & ( ( c_2Etoto_2Eqk__numOrd @ ( c_2Earithmetic_2EBIT1 @ V0x ) @ c_2Earithmetic_2EZERO )
        = c_2EternaryComparisons_2EGREATER )
      & ( ( c_2Etoto_2Eqk__numOrd @ ( c_2Earithmetic_2EBIT2 @ V0x ) @ c_2Earithmetic_2EZERO )
        = c_2EternaryComparisons_2EGREATER )
      & ( ( c_2Etoto_2Eqk__numOrd @ ( c_2Earithmetic_2EBIT1 @ V0x ) @ ( c_2Earithmetic_2EBIT1 @ V1y ) )
        = ( c_2Etoto_2Eqk__numOrd @ V0x @ V1y ) )
      & ( ( c_2Etoto_2Eqk__numOrd @ ( c_2Earithmetic_2EBIT2 @ V0x ) @ ( c_2Earithmetic_2EBIT2 @ V1y ) )
        = ( c_2Etoto_2Eqk__numOrd @ V0x @ V1y ) )
      & ( ( c_2Etoto_2Eqk__numOrd @ ( c_2Earithmetic_2EBIT1 @ V0x ) @ ( c_2Earithmetic_2EBIT2 @ V1y ) )
        = c_2EternaryComparisons_2ELESS )
      & ( ( c_2Etoto_2Eqk__numOrd @ ( c_2Earithmetic_2EBIT2 @ V0x ) @ ( c_2Earithmetic_2EBIT1 @ V1y ) )
        = c_2EternaryComparisons_2EGREATER ) ) ).

thf(thm_2Etoto_2Eap__qk__numto__thm,axiom,
    ( ( c_2Etoto_2Eapto @ tyop_2Enum_2Enum @ c_2Etoto_2Eqk__numto )
    = c_2Etoto_2Eqk__numOrd ) ).

thf(thm_2Etoto_2ETO__charOrd,axiom,
    c_2Etoto_2ETotOrd @ tyop_2Estring_2Echar @ c_2Etoto_2EcharOrd ).

thf(thm_2Etoto_2Eapcharto__thm,axiom,
    ( ( c_2Etoto_2Eapto @ tyop_2Estring_2Echar @ c_2Etoto_2Echarto )
    = c_2Etoto_2EcharOrd ) ).

thf(thm_2Etoto_2EcharOrd__lt__lem,axiom,
    ! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
      ( ( ( c_2Etoto_2EnumOrd @ V0a @ V1b )
        = c_2EternaryComparisons_2ELESS )
     => ( ( ( c_2Eprim__rec_2E_3C @ V1b @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
          = c_2Ebool_2ET )
       => ( ( c_2Etoto_2EcharOrd @ ( c_2Estring_2ECHR @ V0a ) @ ( c_2Estring_2ECHR @ V1b ) )
          = c_2EternaryComparisons_2ELESS ) ) ) ).

thf(thm_2Etoto_2EcharOrd__gt__lem,axiom,
    ! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
      ( ( ( c_2Etoto_2EnumOrd @ V0a @ V1b )
        = c_2EternaryComparisons_2EGREATER )
     => ( ( ( c_2Eprim__rec_2E_3C @ V0a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
          = c_2Ebool_2ET )
       => ( ( c_2Etoto_2EcharOrd @ ( c_2Estring_2ECHR @ V0a ) @ ( c_2Estring_2ECHR @ V1b ) )
          = c_2EternaryComparisons_2EGREATER ) ) ) ).

thf(thm_2Etoto_2EcharOrd__eq__lem,axiom,
    ! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum] :
      ( ( ( c_2Etoto_2EnumOrd @ V0a @ V1b )
        = c_2EternaryComparisons_2EEQUAL )
     => ( ( c_2Etoto_2EcharOrd @ ( c_2Estring_2ECHR @ V0a ) @ ( c_2Estring_2ECHR @ V1b ) )
        = c_2EternaryComparisons_2EEQUAL ) ) ).

thf(thm_2Etoto_2EcharOrd__thm,axiom,
    ( c_2Etoto_2EcharOrd
    = ( c_2Etoto_2ETO__of__LinearOrder @ tyop_2Estring_2Echar @ c_2Estring_2Echar__lt ) ) ).

thf(thm_2Etoto_2Elistorder__ind,axiom,
    ! [A_27a: $tType,V0P: ( A_27a > A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > $o] :
      ( ( ! [V1V: A_27a > A_27a > $o,V2l: tyop_2Elist_2Elist @ A_27a] : ( V0P @ V1V @ V2l @ ( c_2Elist_2ENIL @ A_27a ) )
        & ! [V3V: A_27a > A_27a > $o,V4s: A_27a,V5m: tyop_2Elist_2Elist @ A_27a] : ( V0P @ V3V @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ECONS @ A_27a @ V4s @ V5m ) )
        & ! [V6V: A_27a > A_27a > $o,V7r: A_27a,V8l: tyop_2Elist_2Elist @ A_27a,V9s: A_27a,V10m: tyop_2Elist_2Elist @ A_27a] :
            ( ( V0P @ V6V @ V8l @ V10m )
           => ( V0P @ V6V @ ( c_2Elist_2ECONS @ A_27a @ V7r @ V8l ) @ ( c_2Elist_2ECONS @ A_27a @ V9s @ V10m ) ) ) )
     => ! [V11v: A_27a > A_27a > $o,V12v1: tyop_2Elist_2Elist @ A_27a,V13v2: tyop_2Elist_2Elist @ A_27a] : ( V0P @ V11v @ V12v1 @ V13v2 ) ) ).

thf(thm_2Etoto_2Elistorder,axiom,
    ! [A_27a: $tType] :
      ( ! [V0l: tyop_2Elist_2Elist @ A_27a,V1V: A_27a > A_27a > $o] :
          ( ( c_2Etoto_2Elistorder @ A_27a @ V1V @ V0l @ ( c_2Elist_2ENIL @ A_27a ) )
          = c_2Ebool_2EF )
      & ! [V2s: A_27a,V3m: tyop_2Elist_2Elist @ A_27a,V4V: A_27a > A_27a > $o] :
          ( ( c_2Etoto_2Elistorder @ A_27a @ V4V @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ECONS @ A_27a @ V2s @ V3m ) )
          = c_2Ebool_2ET )
      & ! [V5s: A_27a,V6r: A_27a,V7m: tyop_2Elist_2Elist @ A_27a,V8l: tyop_2Elist_2Elist @ A_27a,V9V: A_27a > A_27a > $o] :
          ( ( c_2Etoto_2Elistorder @ A_27a @ V9V @ ( c_2Elist_2ECONS @ A_27a @ V6r @ V8l ) @ ( c_2Elist_2ECONS @ A_27a @ V5s @ V7m ) )
        <=> ( ( V9V @ V6r @ V5s )
            | ( ( V6r = V5s )
              & ( c_2Etoto_2Elistorder @ A_27a @ V9V @ V8l @ V7m ) ) ) ) ) ).

thf(thm_2Etoto_2ESLO__listorder,axiom,
    ! [A_27a: $tType,V0V: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0V )
     => ( c_2Erelation_2EStrongLinearOrder @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2Elistorder @ A_27a @ V0V ) ) ) ).

thf(thm_2Etoto_2ETO__ListOrd,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] : ( c_2Etoto_2ETotOrd @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2EListOrd @ A_27a @ V0c ) ) ).

thf(thm_2Etoto_2EListOrd__THM,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
      ( ( ( c_2Etoto_2EListOrd @ A_27a @ V0c @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) )
        = c_2EternaryComparisons_2EEQUAL )
      & ! [V1b: A_27a,V2y: tyop_2Elist_2Elist @ A_27a] :
          ( ( c_2Etoto_2EListOrd @ A_27a @ V0c @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ECONS @ A_27a @ V1b @ V2y ) )
          = c_2EternaryComparisons_2ELESS )
      & ! [V3a: A_27a,V4x: tyop_2Elist_2Elist @ A_27a] :
          ( ( c_2Etoto_2EListOrd @ A_27a @ V0c @ ( c_2Elist_2ECONS @ A_27a @ V3a @ V4x ) @ ( c_2Elist_2ENIL @ A_27a ) )
          = c_2EternaryComparisons_2EGREATER )
      & ! [V5a: A_27a,V6x: tyop_2Elist_2Elist @ A_27a,V7b: A_27a,V8y: tyop_2Elist_2Elist @ A_27a] :
          ( ( c_2Etoto_2EListOrd @ A_27a @ V0c @ ( c_2Elist_2ECONS @ A_27a @ V5a @ V6x ) @ ( c_2Elist_2ECONS @ A_27a @ V7b @ V8y ) )
          = ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2Eapto @ A_27a @ V0c @ V5a @ V7b ) @ c_2EternaryComparisons_2ELESS @ ( c_2Etoto_2EListOrd @ A_27a @ V0c @ V6x @ V8y ) @ c_2EternaryComparisons_2EGREATER ) ) ) ).

thf(thm_2Etoto_2Eaplistoto,axiom,
    ! [A_27a: $tType,V0c: tyop_2Etoto_2Etoto @ A_27a] :
      ( ( ( c_2Etoto_2Eapto @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2Elistoto @ A_27a @ V0c ) @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) )
        = c_2EternaryComparisons_2EEQUAL )
      & ! [V1b: A_27a,V2y: tyop_2Elist_2Elist @ A_27a] :
          ( ( c_2Etoto_2Eapto @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2Elistoto @ A_27a @ V0c ) @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Elist_2ECONS @ A_27a @ V1b @ V2y ) )
          = c_2EternaryComparisons_2ELESS )
      & ! [V3a: A_27a,V4x: tyop_2Elist_2Elist @ A_27a] :
          ( ( c_2Etoto_2Eapto @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2Elistoto @ A_27a @ V0c ) @ ( c_2Elist_2ECONS @ A_27a @ V3a @ V4x ) @ ( c_2Elist_2ENIL @ A_27a ) )
          = c_2EternaryComparisons_2EGREATER )
      & ! [V5a: A_27a,V6x: tyop_2Elist_2Elist @ A_27a,V7b: A_27a,V8y: tyop_2Elist_2Elist @ A_27a] :
          ( ( c_2Etoto_2Eapto @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2Elistoto @ A_27a @ V0c ) @ ( c_2Elist_2ECONS @ A_27a @ V5a @ V6x ) @ ( c_2Elist_2ECONS @ A_27a @ V7b @ V8y ) )
          = ( c_2EternaryComparisons_2Eordering__CASE @ tyop_2EternaryComparisons_2Eordering @ ( c_2Etoto_2Eapto @ A_27a @ V0c @ V5a @ V7b ) @ c_2EternaryComparisons_2ELESS @ ( c_2Etoto_2Eapto @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Etoto_2Elistoto @ A_27a @ V0c ) @ V6x @ V8y ) @ c_2EternaryComparisons_2EGREATER ) ) ) ).

thf(thm_2Etoto_2ETO__injection,axiom,
    ! [A_27c: $tType,A_27d: $tType,V0cp: A_27c > A_27c > tyop_2EternaryComparisons_2Eordering] :
      ( ( c_2Etoto_2ETotOrd @ A_27c @ V0cp )
     => ! [V1f: A_27d > A_27c] :
          ( ( c_2Ebool_2EONE__ONE @ A_27d @ A_27c @ V1f )
         => ( c_2Etoto_2ETotOrd @ A_27d @ ( c_2Etoto_2EimageOrd @ A_27d @ A_27c @ V1f @ V0cp ) ) ) ) ).

thf(thm_2Etoto_2EStrongLinearOrder__of__TO__TO__of__LinearOrder,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
      ( ( c_2Erelation_2Eirreflexive @ A_27a @ V0R )
     => ( ( c_2Etoto_2EStrongLinearOrder__of__TO @ A_27a @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0R ) )
        = V0R ) ) ).

thf(thm_2Etoto_2ETO__of__LinearOrder__LEX,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1V: A_27b > A_27b > $o] :
      ( ( ( c_2Erelation_2Eirreflexive @ A_27a @ V0R )
        & ( c_2Erelation_2Eirreflexive @ A_27b @ V1V ) )
     => ( ( c_2Etoto_2ETO__of__LinearOrder @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2ELEX @ A_27a @ A_27b @ V0R @ V1V ) )
        = ( c_2Etoto_2ElexTO @ A_27a @ A_27b @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27a @ V0R ) @ ( c_2Etoto_2ETO__of__LinearOrder @ A_27b @ V1V ) ) ) ) ).

%------------------------------------------------------------------------------